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i∈∆
P
i
∈ T.
Then (P, T) is called a subspace topology of (X, τ).
Theorem 3.2. Let τ be a topology of IFRSs in X and let P ∈ C
X
. Then τ
1
= {P ∩ R :
R ∈ τ} is a topology on P.
The proof is straightforward.
Every member of τ
1
is called open IFRS in (P, τ
1
). If Q ∈ τ
1
, then Q

) ∪S : S ∈ τ
1
} ∪{0
∗
} form a topology of IFRSs on
P of which C
1
is a family of closed IFRSs. But C
1
is also a family of closed IFRSs in (P, τ
1
).
Thus ∃ two topologies of IFRSs τ
1
and τ
2
on P. τ
1
is called the ﬁrst subspace topology of
(X, τ) on P and τ
2
is called the second subspace topology of (X, τ) on P. We brieﬂy write, τ
1
and τ
2
are ﬁrst and second topologies respectively on P, where there is no confusion about the
topological space (X, τ) of IFRSs.
Note 3.5. Let P ∈ C
X
and τ
1
, τ
2
be respectively ﬁrst and second topologies on P. i.e.,
τ
1
= {P ∩ R : R ∈ τ}
and
τ
2
= {(P ∩ P

Department of Mathematics, Acharya Institute of Technology,
Soldevanahalli, Bangalore, 560090, India
E-mail: pskreddy@acharya.ac.in rajanna@acharya.ac.in
Abstract An n-tuple (a
1
, a
2
, ..., a
n
) is symmetric, if a
k
= a
n−k+1
, 1 ≤ k ≤ n. Let H
n
=
{(a
1
, a
2
, ..., a
n
) : a
k
∈ {+, −}, a
k
= a
n−k+1
, 1 ≤ k ≤ n} be the set of all symmetric n-
tuples. A symmetric n-sigraph (symmetric n-marked graph) is an ordered pair S
n
= (G, σ)
(S
n
= (G, µ)), where G = (V, E) is a graph called the underlying graph of S
n
and σ :
E → H
n
(µ : V → H
n
) is a function. In this paper, we deﬁne the antipodal symmetric n-
sigraph A(S
n
) = (A(G), σ) of a given symmetric n-sigraph S
n
= (G, σ) and oﬀer a structural
characterization of antipodal symmetric n-sigraphs. Further, we characterize symmetric n-
sigraphs S
n
for which S
n
∼ A(S
n
) and S
n
∼ A(S
n
) where ∼ denotes switching equivalence and
A(S
n
) and S
n
are denotes the antipodal symmetric n-sigraph and complementary symmetric
n-sigraph of S
n
respectively.
Keywords Symmetric n-sigraphs, symmetric n-marked graphs, balance, switching, antipod
-al symmetric n-sigraphs, complementation.
§1. Introduction
Unless mentioned or deﬁned otherwise, for all terminology and notion in graph theory the
reader is refer to [3]. We consider only ﬁnite, simple graphs free from self-loops.
Let n ≥ 1 be an integer. An n-tuple (a
1
, a
2
, ..., a
n
) is symmetric, if a
k
= a
n−k+1
, 1 ≤ k ≤ n.
Let H
n
= {(a
1
, a
2
, ..., a
n
) : a
k
∈ {+, −}, a
k
= a
n−k+1
, 1 ≤ k ≤ n} be the set of all symmetric
n-tuples. Note that H
n
is a group under coordinate wise multiplication, and the order of H
n
is
2
m
, where m =
n
2
.
A symmetric n-sigraph (symmetric n-marked graph) is an ordered pair S
n
= (G, σ) (S
n
=
(G, µ)), where G = (V, E) is a graph called the underlying graph of S
n
and σ : E → H
n
(µ : V → H
n
) is a function.
In this paper by an n-tuple/n-sigraph/n-marked graph we always mean a symmetric n-
tuple/symmetric n-sigraph/symmetric n-marked graph.
An n-tuple (a
1
, a
2
, ..., a
n
) is the identity n-tuple, if a
k
= +, for 1 ≤ k ≤ n, otherwise it is
a non-identity n-tuple. In an n-sigraph S
n
= (G, σ) an edge labelled with the identity n-tuple
is called an identity edge, otherwise it is a non-identity edge.
Vol. 7 Switching equivalence in symmetric n-sigraphs-IV 35
Further, in an n-sigraph S
n
= (G, σ), for any A ⊆ E(G) the n-tuple σ(A) is the product
of the n-tuples on the edges of A.
In [8], the authors deﬁned two notions of balance in n-sigraph S
n
= (G, σ) as follows (See
also R. Rangarajan and P. S. K. Reddy
[5]
):
Deﬁnition. Let S
n
= (G, σ) be an n-sigraph. Then,
(i) S
n
is identity balanced (or i-balanced), if product of n-tuples on each cycle of S
n
is the
identity n-tuple, and
(ii) S
n
is balanced, if every cycle in S
n
contains an even number of non-identity edges.
Note. An i-balanced n-sigraph need not be balanced and conversely.
The following characterization of i-balanced n-sigraphs is obtained in [8].
Proposition 1.1. (E. Sampathkumar et al.
[8]
) An n-sigraph S
n
= (G, σ) is i-balanced
if, and only if, it is possible to assign n-tuples to its vertices such that the n-tuple of each edge
uv is equal to the product of the n-tuples of u and v.
Let S
n
= (G, σ) be an n-sigraph. Consider the n-marking µ on vertices of S
n
deﬁned as
follows: each vertex v ∈ V , µ(v) is the n-tuple which is the product of the n-tuples on the
edges incident with v. Complement of S
n
is an n-sigraph S
n
= (G, σ
c
), where for any edge
e = uv ∈ G, σ
c
(uv) = µ(u)µ(v). Clearly, S
n
as deﬁned here is an i-balanced n-sigraph due to
Proposition 1.1.
In [8], the authors also have deﬁned switching and cycle isomorphism of an n-sigraph
S
n
= (G, σ) as follows: (See also [4,6,7,11-17])
Let S
n
= (G, σ) and S

n
= (G

, σ

), be two n-sigraphs. Then S
n
and S

n
are said to be
isomorphic, if there exists an isomorphism φ : G → G

such that if uv is an edge in S
n
with
label (a
1
, a
2
, ..., a
n
) then φ(u)φ(v) is an edge in S

n
with label (a
1
, a
2
, ..., a
n
).
Given an n-marking µ of an n-sigraph S
n
= (G, σ), switching S
n
with respect to µ is
the operation of changing the n-tuple of every edge uv of S
n
by µ(u)σ(uv)µ(v). The n-sigraph
obtained in this way is denoted by S
µ
(S
n
) and is called the µ-switched n-sigraph or just switched
n-sigraph.
Further, an n-sigraph S
n
switches to n-sigraph S

n
(or that they are switching equivalent
to each other), written as S
n
∼ S

) are said to be cycle isomorphic, if there
exists an isomorphism φ : G → G

such that the n-tuple σ(C) of every cycle C in S
n
equals to
the n-tuple σ(φ(C)) in S

n
.
We make use of the following known result.
Proposition 1.2. (E. Sampathkumar et al.
[8]
) Given a graph G, any two n-sigraphs
with G as underlying graph are switching equivalent if, and only if, they are cycle isomorphic.
§2. Antipodal n-sigraphs
Singleton
[10]
has introduced the concept of antipodal graph of a graph G as the graph
A(G) having the same vertex set as that of G and two vertices are adjacent if they are at a
distance of diam(G) in G.
36 P. Siva Kota Reddy, M. C. Geetha and K. R. Rajanna No. 3
Motivated by the existing deﬁnition of complement of an n-sigraph, we extend the notion
of antipodal graphs to n-sigraphs as follows: The antipodal n-sigraph A(S
n
) of an n-sigraph
S
n
= (G, σ) is an n-sigraph whose underlying graph is A(G) and the n-tuple of any edge
uv in A(S
n
) is µ(u)µ(v), where µ is the canonical n-marking of S
n
. Further, an n-sigraph
S
n
= (G, σ) is called antipodal n-sigraph, if S
n
∼
= A(S

n
) for some n-sigraph S

n
. The following
result indicates the limitations of the notion A(S
n
) as introduced above, since the entire class
of i-unbalanced n-sigraphs is forbidden to be antipodal n-sigraphs.
Proposition 2.1. For any n-sigraph S
n
= (G, σ), its antipodal n-sigraph A(S
n
) is i-
balanced.
Proof. Since the n-tuple of any edge uv in A(S
n
) is µ(u)µ(v), where µ is the canonical
n-marking of S
n
, by Proposition 1.1, A(S
n
) is i-balanced.
For any positive integer k, the k
th
iterated antipodal n-sigraph A(S
n
) of S
n
is deﬁned as
follows:
A
0
(S
n
) = S
n
, A
k
(S
n
) = A(A
k−1
(S
n
)).
Corollary 2.2. For any n-sigraph S
n
= (G, σ) and any positive integer k, A
k
(S
n
) is
i-balanced.
In [1], the authors characterized those graphs that are isomorphic to their antipodal graphs.
Proposition 2.3. For a graph G = (V, E), G
∼
= A(G) if and only if G is complete.
We now characterize the n-sigraphs that are switching equivalent to their antipodal n-
sigraphs.
Proposition 2.4. For any n-sigraph S
n
= (G, σ), S
n
∼ A(S
n
) if, and only if, G = K
p
and
S
n
is i-balanced signed graph.
Proof. Suppose S
n
∼ A(S
n
). This implies, G
∼
= A(G) and hence G is K
p
. Now, if S
n
is
any n-sigraph with underlying graph as K
p
, Proposition 2.1 implies that A(S
n
) is i-balanced
and hence if S
n
is i-unbalanced and its A(S
n
) being i-balanced can not be switching equivalent
to S
n
in accordance with Proposition 1.2. Therefore, S
n
must be i-balanced.
Conversely, suppose that S
n
is an i-balanced n-sigraph and G is K
p
. Then, since A(S
n
)
is i-balanced as per Proposition 3 and since G
∼
= A(G), the result follows from Proposition 2
again.
Proposition 2.5. For any two n-sigraphs S
n
and S

n
with the same underlying graph,
their antipodal n-sigraphs are switching equivalent.
Proposition 2.6. (Aravamudhan and Rajendran
[1]
) For a graph G = (V, E),
G
∼
= A(G) if, and only if, i). G is diameter 2 or ii). G is disconnected and the components of
G are complete graphs.
In view of the above, we have the following result for n-sigraphs:
Proposition 2.7. For any n-sigraph S
n
= (G, σ), S
n
∼ A(S
n
) if, and only if, G satisﬁes
conditions of Proposition 2.6.
Proof. Suppose that A(S
n
) ∼ S
n
. Then clearly we have A(G)
∼
= G and hence G satisﬁes
conditions of Proposition 2.6.
Conversely, suppose that G satisﬁes conditions of Proposition 2.6. Then G
∼
= A(G) by
Proposition 2.6. Now, if S
n
is an n-sigraph with underlying graph satisﬁes conditions of Propo-
Vol. 7 Switching equivalence in symmetric n-sigraphs-IV 37
sition 2.6, by deﬁnition of complementary n-sigraph and Proposition 2.1, S
n
and A(S
n
) are
i-balanced and hence, the result follows from Proposition 1.2.
The following result characterize n-sigraphs which are antipodal n-sigraphs.
Proposition 2.8. An n-sigraph S
n
= (G, σ) is an antipodal n-sigraph if, and only if, S
n
is i-balanced n-sigraph and its underlying graph G is an antipodal graph.
Proof. Suppose that S
n
is i-balanced and G is a A(G). Then there exists a graph H
such that A(H)
∼
= G. Since S
n
is i-balanced, by Proposition 1.1, there exists an n-marking
µ of G such that each edge uv in S
n
satisﬁes σ(uv) = µ(u)µ(v). Now consider the n-sigraph
S

= (H, σ

), where for any edge e in H, σ

(e) is the n-marking of the corresponding vertex in
G. Then clearly, A(S

i=∞
i=1
λ
i
where each λ
i
∈ T.
The complement of a fuzzy G
δ
is a fuzzy F
σ
set.
Proposition 2.1.
[5]
For any IFS A in (X, τ) we have
(i) cl(A) = int(A).
(ii) int(A) = cl(A).
Deﬁnition 2.7.
[6,8]
Let A be an IFS of an IFTS X. Then A is called an
(i) intuitionistic fuzzy regular open set (IFROS) if A = int(cl(A)).
(ii) intuitionistic fuzzy semi open set (IFSOS) if A ⊆ cl(int(A)).
(iii) intuitionistic fuzzy β open set (IFβOS) if A ⊆ cl(int(cl(A))).
(iv) intuitionistic fuzzy preopen set (IFPOS) if A ⊆ int(cl(A)).
(v) intuitionistic fuzzy α open set (IFαOS) if A ⊆ int(cl(int(A))).
Deﬁnition 2.8.
[6,8]
Let A be an IFS in IFTS. Then
(i) βint(A) = ∪{G/G is an IFβOS in X and G ⊆ A} is called an intuitionistic fuzzy
β-interior of A.
(ii) βcl(A) = ∩{G/G is an IFβCS in X and G ⊇ A} is called an intuitionistic fuzzy β
-closure of A.
(iii) int
s
(A) = ∪{G/G is an IFSOS in X and G ⊆ A} is called an intuitionistic fuzzy semi
interior of A.
Vol. 7 A view on intuitionistic fuzzy regular G
δ
set 79
(iv) cl
s
(A) = ∩{G/G is an IFCS in X and G ⊇ A} is called an intuitionistic fuzzy semi
closure of A.
Deﬁnition 2.9.
[9]
A fuzzy topological space (X, T) is said to be fuzzy β − T
1/2
space if
every gfβ-closed set in (X, T) is fuzzy closed in (X, T).
§3. Intuitionistic fuzzy regular G
δ
set and it’s properties
Deﬁnition 3.1. Let (X, T) be an intuitionistic fuzzy topological space. Let A = {x, µ
A
(x),
γ
A
(x) : x ∈ X} be an intuitionstic fuzzy set of an intuitionistic fuzzy topological space X.
Then A is said to be an intuitionistic fuzzy G
δ
set (inshort, IFG
δ
S) if A =

α∈Γ
A
α
is a fuzzy
super b-connected subset of X.
Proposition 2.12. Let (X, T) be a smooth fuzzy topological space. Suppose that X is
fuzzy super b-connected and C is a fuzzy super b-connected subset of X. Further, suppose that
X\C contains a set V such that 1
V
/X\C is a r-fuzzy b-open set in the fuzzy subspace X\C of
X. Then C